Originally Posted by **avito009** So E is the hypotenuse and the base is pc. So base cannot be more than hypotenuse. So let us assume E= pc. so after substituting mc2 in E which gives us
mc2=pc now substitute mv into p this gives us
mc2= mvc after cancelling what remains is v=c. But we know that velocity of an object cant be equal speed of light. so E is not equal to pc.
But in that case E=pc in case of light?
Am i right? |

In terms of high speeds near speed of light we need to take into account relativistic mass so $\displaystyle p = mv$ becomes $\displaystyle p = \frac{ m_{0}v}{\sqrt{1 - \frac{v^{2}}{c^{2}}}}$ where m0 is rest mass. If you are talking about massless particles like photons then it is correct to use E=pc. Otherwise, if the particle has mass, then $\displaystyle E=m c^2$ becomes $\displaystyle E=\frac{m_{0} c^{2}}{\sqrt{1 - \frac{v^{2}}{c^{2}}}}$ and the m0 is again rest mass moving at velocity v. (Note: The E we are talking about here is kinetic, not potential energy.)

Now for low speeds much slower than light we can approximate $\displaystyle E=\frac{m_{0} c^{2}}{\sqrt{1 - \frac{v^{2}}{c^{2}}}} \approx \frac{m_{0} v^{2}}{2} + m_{0} c^{2} + \mathcal{O}\left(v^{4}\right)$ using Taylor's series expansion. This shows that for low speeds $\displaystyle \frac{m_{0} v^{2}}{2}$ is a good approximation of kinetic energy.

Hence for low speeds and for ordinary physical conditions like dropping a ball off a cliff then using $\displaystyle E_{kinetic} = \frac{m_{0} v^{2}}{2}$ with $\displaystyle E_{potential} = m g h$ along with conservation of energy principles makes sense. However. trying to use relativistic formulae in such cases is "overkill" and is a pain to calculate minor increases in inertial mass for low velocity when $\displaystyle \frac{m_{0} v^{2}}{2}$ does just as well. So remember when you see E=mc^2 it is not a formula about rest mass (m0) but the inertial mass ($\displaystyle m = \frac{m_{0}}{\sqrt{1 - \frac{v^{2}}{c^{2}}}}$ ) which is not a constant for relativistic velocities and is generally not a formula we would use for low velocities when the difference between rest mass and inertial mass are negligible.